Auction Design without Commitment



Feasible choice rules To highlight the fact that in the above definition the
English auction is conditioned on the prior
p, denote it by φE [p]. Now the function
φE : ∆(Θ) → Φ can be taken as the the English auction -choice rule. Construct a
correspondence
CφE VIC such that

Cφ [p] = ©Ф VIC[p] : Ф is not ex post dominated by φE[p0], for any p' ∆(Θ)} .

Note that the English auction -choice rule is idempotent: φE [p(x, φE [p])] =
1x , for all x φE [p](supp(p)), for all p. That is, after running the English auction,
a new English auction does not change the outcome. This implies that the English
auction
φE [p] is not ex post dominated by φE[p(x, φE [p])]. Hence it follows that
φE [p] CφE [p] for all p. More compactly:

Lemma 3 φE [∙] is consistent.

If φE [p] also maximizes v in CφE [p] for all p, then φE satisfies the one-deviation
property. We now show that this necessarily holds:
any mechanism in CφE [p] is
outcome equivalent - and hence payoff equivalent - with the English auction
φE [p].

Lemma 4 φ CφE [p] only if φ is outcome equivalent to φE [p], for all p.

Proof. Relegated to Appendix A. ■

That is, given p, the only veto-incentive compatible mechanisms that are not ex
post dominated by any
φE [p0] are the mechanism φE [p] itself and its versions that
may additionally reveal some non-relevant information concerning the winner’s
type
. Thus φE has a "fixed point" property.

We next demonstrate that any stationary σ that is consistent and meets
the one-deviation property induces
Cσ that contains φE , specified for some tie-
breaking rule
w. Heuristically, if φ ex post dominates φE [p], then φ must change
φE [p]’s allocation. Since any outcome x of φE [p] reveals the winner and his least
possible valuation given the other buyers’ valuations,
φ must threaten the win-
ner to sell the good to the buyer with the second highest valuation to force the
winner to pay a higher price. However, the threat is not credible since when the
winner declines the offer, the seller sells, by stationarity, to the winner with his
least possible valuation.

Lemma 5 Let a stationary choice rule σ be consistent and satisfy the one-deviation
property. Then there is a tie-breaking rule
w such that φE [p] Cσ [p], for all p.

Proof. Construct w as follows: For any θ Θ, denote by 1θ the degenerate
prior such that supp
(1θ) = {θ}. Then there is an outcome xθ such that σ[1θ] = 1xθ .
Let w(θ)=i if xθ allocates the good to i. By Lemma 2, such w(θ) satisfies (5).
Use this
w to construct φE .

Suppose, to the contrary of the claim, that there is p such that φE [p] 6 Cσ [p].
Then
φE[p] is ex post dominated by σ[p(x, φE [p])] for some x φE[p](supp(p)).
Denote σ[p(x, φE [p])] = g h.

12



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